Minimum XOR Path in a Grid - Solution & Explanation

Problem Overview: Given a grid of integers, move from the top-left cell to the bottom-right cell while accumulating the XOR of all visited values. The goal is to minimize the final XOR value. Movement is typically restricted to right and down, which means every path has a fixed length but different XOR outcomes.

Approach 1: Brute Force DFS Enumeration (Exponential Time)

The most direct strategy is to enumerate every valid path from (0,0) to (m-1,n-1). During the DFS traversal, keep a running XOR of the values along the path. When you reach the destination, update the minimum XOR seen so far. This explores roughly 2^(m+n) paths in the worst case because each step branches into two directions. Time complexity is O(2^(m+n)) and space complexity is O(m+n) due to recursion depth. This approach works only for very small grids but helps verify correctness and understand the XOR accumulation.

Approach 2: Dynamic Programming with XOR State Sets (O(m * n * K))

A more structured method stores all reachable XOR values at each cell. Define dp[i][j] as the set of XOR values obtainable when reaching cell (i,j). For each previous state from (i-1,j) or (i,j-1), compute newXor = prevXor ^ grid[i][j] and insert it into the current set. This effectively tracks all possible XOR results along different paths. If the value range is bounded, the number of XOR states per cell remains manageable. Time complexity becomes roughly O(m * n * K), where K is the number of distinct XOR states, and space complexity is O(m * n * K). This approach is a natural extension of grid dynamic programming with bit manipulation.

Approach 3: Meet-in-the-Middle Path XOR (Optimized)

For larger grids, enumerating full paths becomes expensive. Instead, split the path length in half. Generate all partial paths from the start until the middle layer and store their XOR values in hash maps keyed by position. Then generate paths backward from the destination to the same middle layer and combine XOR results. The final XOR of a complete path equals leftXor ^ rightXor ^ grid[mid], so checking combinations allows you to track the minimum result efficiently. Each side enumerates about half the path depth, reducing complexity from exponential in m+n to roughly O(2^((m+n)/2)) time and space. This technique appears often in graph traversal problems where path enumeration is unavoidable.

Recommended for interviews: Start by explaining the brute force DFS to show how XOR accumulates along a path. Then transition to the state-based DP or meet-in-the-middle optimization. Interviewers usually expect you to recognize that enumerating full paths is exponential and reduce the search space using XOR state tracking or path splitting.